Skip to main content
SpringerLink
Log in

The existence of cyclic (v, 4, 1)-designs

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Even though Peltesohn proved that a cyclic (v, 3, 1)-design exists if and only if \(v\equiv 1,3\pmod {6}\) as early as 1939, the problem of determining the spectrum of cyclic (v, k, 1)-designs with \(k>3\) is far from being settled, even for \(k=4\). This paper shows that a cyclic (v, 4, 1)-design exists if and only if \(v\equiv 1,4\pmod {12}\) and \(v\not \in \{16,25,28\}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abel R.J.R., Buratti M.: Some progress on \((v,4,1)\) difference families and optical orthogonal codes. J. Comb. Theory A 106, 59–75 (2004).

    Article  MathSciNet  Google Scholar 

  2. Abel R.J.R., Buratti M.: Difference families. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs (2nd Edition), pp. 392–410. CRC Press, Boca Raton (2007).

    Google Scholar 

  3. Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugen. 9, 353–399 (1939).

    Article  MathSciNet  Google Scholar 

  4. Buratti M.: Improving two theorems of Bose on difference families. J. Comb. Des. 3, 15–24 (1995).

    Article  MathSciNet  Google Scholar 

  5. Buratti M.: Constructions for \((q, k,1)\) difference families with \(q\) a prime power and \(k=4,5\). Discret. Math. 138, 169–175 (1995).

    Article  MathSciNet  Google Scholar 

  6. Buratti M.: From a \((G, k,1)\) to a \((C_k\oplus G, k,1)\) difference family. Des. Codes Cryptogr. 11, 5–9 (1997).

    Article  MathSciNet  Google Scholar 

  7. Buratti M.: Recursive constructions for difference matrices and relative difference families. J. Comb. Des. 6, 165–182 (1998).

    Article  MathSciNet  Google Scholar 

  8. Buratti M.: Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111–125 (2002).

    Article  MathSciNet  Google Scholar 

  9. Buratti M., Pasotti A.: Further progress on difference families with block size \(4\) or \(5\). Des. Codes Cryptogr. 56, 1–20 (2010).

    Article  MathSciNet  Google Scholar 

  10. Chang Y.: Some cyclic BIBDs with block size four. J. Comb. Des. 12, 177–183 (2004).

    Article  MathSciNet  Google Scholar 

  11. Check P.L., Colbourn C.J.: Concerning difference families with block size four. Discret. Math. 133, 285–289 (1994).

    Article  MathSciNet  Google Scholar 

  12. Chen K., Wei R.: A few more cyclic Steiner 2-designs. Electron. J. Comb. 13, R10, 1–13 (2006).

  13. Chen K., Zhu L.: Existence of \((q, k,1)\) difference families with \(q\) a prime power and \(k=4,5\). J. Comb. Des. 7, 21–30 (1999).

    Article  MathSciNet  Google Scholar 

  14. Chung F.R.K., Salehi J.A., Wei V.K.: Optical orthogonal codes: design, analysis and applications. IEEE Trans. Inf. Theory 35, 595–604 (1989).

    Article  MathSciNet  Google Scholar 

  15. Colbourn M.J., Colbourn C.J.: Recursive constructions for cyclic block designs. J. Stat. Plann. Inference 10, 97–103 (1984).

    Article  MathSciNet  Google Scholar 

  16. Colbourn M.J., Mathon R.A.: On cyclic Steiner 2-designs. Ann. Discret. Math. 7, 215–253 (1980).

    Article  MathSciNet  Google Scholar 

  17. Ge G., Miao Y., Sun X.: Perfect difference families, perfect difference matrices, and related combinatorial structures. J. Comb. Des. 18, 415–449 (2010).

    Article  MathSciNet  Google Scholar 

  18. Hanani H.: The existence and construction of balanced incomplete block designs. Ann. Math. Stat. 32, 361–386 (1961).

    Article  MathSciNet  Google Scholar 

  19. Heffter L.: Über Nachbarconfigurationen, Tripelsysteme und metacyklische Gruppen. Deutsche Mathem. Vereinig. Jahresber. 5, 67–69 (1896).

    MATH  Google Scholar 

  20. Heffter L.: Über Tripelsysteme. Math. Ann. 49, 101–112 (1897).

    Article  MathSciNet  Google Scholar 

  21. Kirkman T.P.: On a problem in combinations. Camb. Dublin Math. J. 2, 191–204 (1847).

    Google Scholar 

  22. Köhler, E.: \(k\)-difference-cycles and the construction of cyclic \(t\)-designs. In: Geometries and Groups. Lecture Notes in Mathematics, vol. 893, pp. 195–203. Springer, Berlin (1981)

  23. Netto E.: Zur Theorie der Tripelsysteme. Math. Ann. 42, 143–152 (1893).

    Article  MathSciNet  Google Scholar 

  24. Peltesohn R.: Eine Lösung der beiden Heffterschen Differenzenprobleme. Compos. Math. 6, 251–257 (1939).

    MathSciNet  MATH  Google Scholar 

  25. Reid C., Rosa A.: Steiner systems \(S(2,4,v)\)—a survey. Electron. J. Comb. DS18 (2010).

  26. The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.11.1, 2021. https://www.gap-system.org.

  27. Yang Y., Lin X.: Coding Theory and Cryptography (in Chinese). Peoples Post and Telecommunications Press, Beijing (1992).

    Google Scholar 

  28. Yin J.: Some combinatorial constructions for optical orthogonal codes. Discret. Math. 185, 201–219 (1998).

    Article  MathSciNet  Google Scholar 

  29. Zhang M., Feng T., Wang X.: The existence of cyclic \((v,4,1)\)-designs (2022). https://doi.org/10.5281/zenodo.6370238.

Download references

Acknowledgements

The authors thank the editor, Marco Buratti, and the anonymous referees for their valuable comments and suggestions that helped improve the equality of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Menglong Zhang & Tao Feng

  2. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Xiaomiao Wang

Authors
  1. Menglong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaomiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Feng.

Additional information

Communicated by M. Buratti.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by NSFC under Grant 11871095 (Tao Feng), NSFC under Grant 11771227 and Zhejiang Provincial Natural Science Foundation of China under Grant LY21A010005 (Xiaomiao Wang)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, M., Feng, T. & Wang, X. The existence of cyclic (v, 4, 1)-designs. Des. Codes Cryptogr. 90, 1611–1628 (2022). https://doi.org/10.1007/s10623-022-01057-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01057-9

Keywords

Mathematics Subject Classification

Search

Navigation